Testing for One-Factor Models versus Stochastic Volatility Models

-Xi∕_n∣<ξn}²ⁿ ʃj/n^/ (X^{s  X}j/n) ^σ(X^s)dW^{1 ,s}

У^n-1 1

2=₃=1 ¹ {∣Xj∕n-x_i∕n<M

S/

Gn,m,r

-X_i∕_n∖<ξn} ^{2 n} ʃj/n )   ( X^s   Xj/n ) ^μ ( X^{s )d s}

∑n-1 ₁

j =1 ¹ {∣Xj∕n-X_i∕n∣<ξn}

SZ

Hn,m,r

-X,,n∖<ξn}ⁿ (ʃj/"¹¹ /n (σ²(Xs) - σ²(Xi/n)) ds

∑n-1

j =1 ¹ {∣Xj∕n-X_i∕n∣<ξn}

--------S/----------------------------

Dn,m,r

Thus, we need to show that G_n,_m,_r, H_n,_m,_r and D_n,_m,_r are o_P(1).
Now, because of Lemma 1,

Dn,m,r ≤ √m   sup    Iσ²(Xs) - σ²(X_τ)∣

∖X_s-X_τ ∣≤ξn

≤ √m sup IVσ²(X_τ)∣   sup   ∖X_s — X_τ∣

τ∈ [0,1]                ∖Xs-Xτ ∣≤ξn

= O (√m) O_a.s. (n^ε/²)O_a.s(ξn) = o_a.s. (1),

provided that m¹ /²n^ε/²ξ_n → 0. Since m = o(n^{1 ε}), then

Oa.s ( √mn^ε/²ξn) = 0a.s (n¹ /²ξn),

which approaches zero almost surely.

As for G_n,_m,_r, by the proof of Step 1 of Theorem 1, part(i)a, below, (√n∕√m)G_n,_m,_r = G_n,_r
converges in distribution and so it’s O_p(1); therefore G_n,_m,_r = o_P(1), given that m∕n → 0, as
m,n → ∞.

Finally, given the continuity of μ(■ ),

H[s_npι,r ∣ ≤ √m sup ∖μ(Xs) ∣    sup    ¹ Xs — Xi/n¹

s∈ [0,1]            |i/n-s∖≤ 1 /n

s∈[0,1]

= √mO_as_i( (n^ε/⁴)Oa.s. (n^-1 /² log n) = oa.s. (1).                  (17)

In fact, because of the modulus of continuity of a diffusion (see McKean, 1969, p.96),

sup    ¹ Xs — Xi/n ¹ = Oa.s. n-~¹ /² logn) ,

[i/n-s^i/n
s∈[0,1]

and n¹ /^2-ε+ε/⁴n^-1 /² log n = n^-3ε/⁴ log n → 0. Therefore, the statement follows.                 ■

We can now prove Theorem 1.

19

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